and \(m_a^{\rm QCD}\) , \(f_a\) , \(f_\pi , m_\pi , m_u\)  and \(m_d\)  denote respectively the axion mass, the axion scale, the pion decay constant, and the pion, up and down quark masses [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. Eq. (REF ) holds whenever QCD is the only confining group of the theory,
irrespectively of the ultraviolet (UV) model details. A departure from this \(m_a\) -\(f_a\)  relation always requires to extend the gauge confining sector beyond the Standard Model (SM) QCD group.
